Solve y^2dx+(xy+x^2)dy=0

$$y ^ { 2 } d x + ( x y + x ^ { 2 } ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=0\text{ or }y=-\frac{x}{2}\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply xy+x^{2} by d.

$$y^{2}dx+\left(xyd+x^{2}d\right)y=0$$

Use the distributive property to multiply xyd+x^{2}d by y.

$$y^{2}dx+xdy^{2}+x^{2}dy=0$$

Combine y^{2}dx and xdy^{2} to get 2y^{2}dx.

$$2y^{2}dx+x^{2}dy=0$$

Combine all terms containing d.

$$\left(2y^{2}x+x^{2}y\right)d=0$$

The equation is in standard form.

$$\left(2xy^{2}+yx^{2}\right)d=0$$

Divide 0 by 2y^{2}x+x^{2}y.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=0\text{ or }y=-\frac{x}{2}\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply xy+x^{2} by d.

$$y^{2}dx+\left(xyd+x^{2}d\right)y=0$$

Use the distributive property to multiply xyd+x^{2}d by y.

$$y^{2}dx+xdy^{2}+x^{2}dy=0$$

Combine y^{2}dx and xdy^{2} to get 2y^{2}dx.

$$2y^{2}dx+x^{2}dy=0$$

Combine all terms containing d.

$$\left(2y^{2}x+x^{2}y\right)d=0$$

The equation is in standard form.

$$\left(2xy^{2}+yx^{2}\right)d=0$$

Divide 0 by 2y^{2}x+x^{2}y.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-2y\text{; }x=0\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=-2y\text{; }x=0\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$